Abstract: Phylogenetic Trees are a fundamental tool for summarising the mutation structure of many diseases. In practice, we cannot identify an exact tree, but instead identify a sample of likely trees given the observed leaf data. Probability metrics are a popular tool for comparing statistical samples, but require our samples to live in the same space; this is generally not the case when studying the phylogenetic trees of independent evolutionary processes. To this end, we define probability metrics between distributions on tree spaces of different dimensions. Using the identification of the tropical Grassmannian and phylogenetic tree space [Speyer and Sturmfels, The Tropical Grassmannian], we study Wasserstein distances between general measures on tropical projective torii. Our method mirrors the semi-orthogonal map methodology used by Cai and Lim [Cai and Lim, Distances Between Probability Distributions of Different Dimensions] in the Euclidean setting, establishing the same powerful behaviour in a more complex tropical setting.